A vast spectrum of modern problems center on understanding the status and dynamical behavior of networks. For example, our entire national and world economic system of financial transfers, shipments of goods, transportation of people, delivery of utilities, and the contagion of disease can only be managed and understood if we are able to understand the behavior of networks. One of the most daunting of these problems is the emergence of the internet for computer and personal communication including the remote control of devices by internet, by software or persons from a distant site often without our knowledge or understanding. Foremost among these problems are the emergence of computer bugs, worms, viruses, attacks, and an entire spectrum of malicious processes requiring something akin to the biological defenses necessary for the maintenance of life forms. All of these systems can be described as set of nodes where some nodes are connected by a weight of connection that can be zero or a positive number and thus constitute a network.
Another type of network that can reek havoc on modern society are networks of both criminal and terrorist groups. If we cannot maintain secure communication and control for our military complex then we become subject to attacks and destruction of our social order to an extent never before conceivable. Thus the problem of understanding, monitoring, tracking, and securing networks is of the greatest possible importance to the security of our nation, world order, and the very survival of advanced civilization.
The problem of understanding networks is thus of the greatest possible importance for the future stability of a complex social system. The central network problem resides in the shear volume of data and the fact that any one value can be of the same importance as every other value. What are needed are systems and methods for determining and utilizing summary statistical data to understand, monitor, and analyze the status and dynamical behavior of networks networks.